On a Better Golden Rectangle (That Is Not 61.8033. . .% Useless!)

Douglas M. McKennaMathemæsthetics, Inc., Boulder, Colorado, USA; doug@mathemaesthetics.com

AbstractIn which (A) it is alliteratively alleged that the φ×1 Golden Rectangle™ is a φction of self-referential spatial elegance,unworthy of the reverent φxation with which φlosophers of beauty have for too long worshipped it; and (B) it isdeφnitively demonstrated that the

√φ × 1 rectangle is a far more satisφing, lean, golden mean, æsthetic φst-φghting

machine. We explore its recursive subdivision into self-tilings exhibiting emergent global structure and patterns, andshow its relationship both to golden ratio-based identities and to Ammann’s bee tile, with a little art phistory added.

Complaint

For centuries, indeed millennia, smart people who should know better have ex-, de-, pro- andgenerally -claimed that the Golden Rectangle™ is fabulous, and wonderful, and oh-so-superblyproportioned, the very ephitome1 of framed elegance among all things artfully rectangular, mostworthy of all manner of æsthetic reference, preference, and deference, yada, yada, yada.

Well, it’s time to put a stop to this facile, faithful, unfulφlling nonsense. Of all things auric, theGolden Rectangle™ is—in my not so humble ophinion—far and away just about the worst exampleof “wrecked-angular” elegance in the parthen. . . er, pantheon of φ-dom. In short, yes, I’m sorry tosay, the over-marketed, putatively sacred Golden Rectangle™ is a mathemæsthetic object unφt forthe burden of elegance and phiness(e) so worshipfully and steamingly heaped upon the poor thing.

As an Allegedly Self-Referential Object, the Golden Rectangle™ Is Nearly 62% Useless!

Phigure 1, left, shows yet another Miserable Golden Rectangle™ (hereinafter MGR) in its custom-arily ordained, “canonical” (ahem) wider-than-high form. Its dimensions are either 1 × φ, whereφ = (−1 +

√5)/2 = 0.618033 . . . , or φ × 1, where φ = (1 +

√5)/2 = 1.618033 . . . . Choose your

x2 − x − 1 = 0 poison; here we will opt for the larger (by) one (φ = 1.618033 . . . ).

1

ƴ

1

1

Useless!

ƴ - 1

Useless!

Use

less

!

Useless!

Useless!

...

Phigure 1: The usual MGR subdivision accumulates self-referentially useless, non-Golden squares.

The salient, semi-self-reverential property that everyone always oohs and aahs about is this: ifyou carve off a square of side 1 from the MGR, you are left with another smaller 1 × 1

φ MGR,

1Yes, my ‘h’s and/or ‘p’s can be silent. For non-English-speaking φlomorphs, feel free to sound-substitute “phi” or “fi” (asappropriate) for ‘φ’.

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rotated 90° (Phigure 1, center). And hence, one can perform another carving on the smallerMGR, iterating ad inφnitum (Phigure 1, right), without drifting away from the same, simply divineproportion, which drifting otherwise occurs when using a greater or (judgmentally speaking) lesserratio. From there—assuming one makes the appropriate binary symmetry choice forever—the usualtrue discussion of logarithmic spirals and false discussion of nautilus shells [3] ensues. Yawn.

Although the MGR is the only rectangle for which this goldenectomy holds—thereby selφshlydrawing much unwarranted attention to itself—the blatant elegance problem is that, after dividingthe MGR into two pieces, the smaller piece (just under 2/5 of the original) is the only part thatis still golden; the majority (just over 3/5) is just . . . a flippin’ (i.e., symmetric) unit square! Butthe unit (or any other) square has absolutely nothing golden about it to recommend itself. Nada.0.0000000. Zip. Zilch. 7

ℵ0. 041. The so-called “empty set.”

⋃∞0

i=0 0 × 0; see, e.g., lost causes.It’s like being presented with a piece of meat, billed as a φne steak, but which is fully 1/φ =

61.8 . . .% inedible fat, to be discarded (or fed to φdo) by those of us who are the more discerningephicures! Any φnagling butcher foisting this scam off onto his customers would be rightfullysubject to complaint, maybe even Madofφan conφnement for a φnancial pyramid2 scheme!

The φ × 1 rectangle is, then, only minimally golden-ish. Its boring and useless square gnomon(that which is left over after removing a self-similar piece) is a symptom of the MGR’s inherentinelegance and diminished mathematical beauty. MGR is, quite literally (well, okay, numerately)an æsthetic misφt, due to a fundamental φ-ological mistake: Everyone’s been shoehorning φ’s one-dimensional wonder into the wrong two-dimensional rectangle! Bear with me now in proving this.

Divi nd ing a Rectangle into Self-Similar Pieces

Putting wordplay, snark, sound puns, and typogra-phi-cal mischief temporarily aside, without lossof congeniality, consider an r × 1 rectangle, where r ≥ 1, shown wider than high. What value(s)of r permit this rectangle to be subdivided into exactly N similar sub-rectangles R1, · · · ,RN ?

The trivial answer, for N =1 and r =1, says that a square is similar to itself. The obvious non-trivial answer, for N =4, is also r =1, i.e., a square can be subdivided into four similar sub-squares.And indeed, for N = k2, any rectangular r works. But what values of r work for N =2 or N =3?

The well-known solution for N = 2 is the√

2 × 1 rectangle, which can be halved into twosimilar sub-rectangles, each rotated 90°. This is the basis of European paper sizes, where, forinstance, a sheet of paper of size A3 can be cut into exactly two sheets of size A4. Subdivide justone of those two sub-rectangles and, by induction, the

√2 × 1 rectangle will work for any N .

As shown in Figure 2, when N = 3 there are three distinct (disregarding symmetry) solutions.With a bit of algebra, one finds the three solutions are r =

√2,√

3, or . . .√φ.

1

2 3

A

ƴ

1

Figure 2: Three distinct solutions to dividing a rectangle into three self-similar smaller rectangles.

2Do not—I repeat, do not!—get me started on MGR and the pyramids, or I will need a deφbrillator!

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The third√φ × 1 = 1.2720196 . . . × 1 solution, which we will call GR, is unexpectedly

interesting, not just because of the appearance of√

the golden ratio φ, but also because each of thethree similar sub-rectangles is a different size, accomplished using different orientations.

Let r =√φ. Scale each of GR’s side lengths by r4 = φ2, so that while remaining similar

it becomes an r5 × r4 rectangle (Figure 3, center). Label the three component rectangles in thesubdivision as A, B, and C, from largest to smallest. Sub-rectangle A is similar, so its short sideis r3 and its area is thus r4r3 = r7. The short side of sub-rectangle B must be (r5 − r3) = r (r4 − r2).But recall that φ2 − φ = 1, i.e., r4 − r2 = 1. So the short side of B simplifies to just r , which bysimilarity implies that B’s long side is r2. But that “in turn” means that sub-rectangle C must ber1 × r0 = r × 1. So we see that r5 = r3 + r (horizontal lengths) and that r4 = r2 + 1 (verticallengths), both of which are the same golden ratio identity as φ2 = φ + 1. And summing areas, weget r9 = r7 + r3 + r , which upon dividing by r is equivalent to the related identity φ4 = φ3 + φ + 1.

ƴr =

1

AB

C r 0= 1

r 2

rr 3

r 4

r 50

0

1

2

3

0

1

2

3

0 1

2 3

0 1

32

Figure 3: Left: The subdivided√φ × 1 rectangle. Middle: Scaled by r4 = φ2. Right: Each of A, B, and C

has four possible coordinate systems based on which corner is treated as an origin.

So GR gets to be goldenly recursive everywhere—no ugly leftover squares that infinitely sum tognomonic uselessness with area

∑∞i=0 φ

−i = 1φ−1 = φ × 1 = MGR (à la Phigure 1, right). QED.

Recursive Subdivision Everywhere Down to a Constant Level

Using translation, mirroring, and rotation, any GR has four possible coordinate systems by whichto anchor a subdivision, where each corner can be an origin, labeled 0–3 (Figure 3, center andright). Therefore, there are 43 = 64 ways to assign coordinate systems to the three component sub-rectangles A, B, and C, given the current coordinate system of their parent GR. Once a coordinatesystem is chosen for each of A, B, and C, one can recursively carve each into a next level of sub-sub-rectangles, which composes the transformations iteratively down to some level. Because thesubdivision into three self-similar rectangles is asymmetric, the 64 possibilities have distinct forms.

The results are visually fascinating: wispy, self-similar cloud-like patterns emerge at manyscales, due to how the tree of linear transformations brings the largest (lightest) As and the smallest(darkest) Cs together spatially in various ways. For a given recursive limit down n levels of A, B,and C, the result for each is a self-similar tapestry of 3n GRs, in a variety of different sizes andorientations. Figure 4 shows four such recursive subdivisions for n = 9 (each rotated 90° to betterfit on the page). Each features distinct, emergent self-similar patterns created by the exact same setof 39 = 19683 GRs. When using the same line width to draw subdivision lines, smaller rectanglesappear darker or merge. Global structure emerges from local texture. Is this not way more beautifulthan one logarithmic spiral, which is self-similar at only one measly 0-dimensional point?

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A3 B

3 C

3

A0 B

1 C

3

A1 B

0 C

2

A2 B

1 C

3

Figure 4: Four of the 64 subdivision symmetries, each n=9 levels deep. By construction, each uses theexact same set of 39 = 19683 sub-tiles. Top left: (A3 B3 C3) A new set of GRs emerges alongrotated lines. Top right: (A0 B1 C3) The smallest GRs coalesce into more sinuous patterns.

Bottom left: (A1 B0 C2) Seemingly chaotic swirls emanating from one corner. Bottom right: (A2B1 C3) A self-referential diagonal sweep.

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Subdivision Lags Result in Aperiodic Tilings Using Only Four Sizes of GR

The golden ratio is a special constant because it represents an exact balancing point betweenadditive and multiplicative growth. Hence, a recursive subdivision of A down more levels than ofsmaller B, and of B down a little more than of smaller C, yields only a small set of sub-rectanglesexactly comparable in size. So at each recursive level n, subdivide A to level n, B to level n − 2,and C to level n − 3. These lags counterbalance the decreasing exponents on the right side ofthe area identity φ4 = φ3 + φ1 + φ0. Each of the 64 “size-limited” results is a tiling of GR withsmaller copies of itself. But only four different tile sizes are produced. If, for each starting level n,GR is scaled by φ, every such tiling for any given n then represents an identity that defines φn asa sum of the first four powers of φ, i.e., φn = aφ3 + bφ2 + cφ + d, where the integer values ofa, b, c, d count the number of tiles of each of the four size classes, in order of decreasing area.By construction, each of the four lag subdivisions in Figures 5 and 6 comprises the exact sameset of tiles. In Figure 6, right, one can easily verify that the lag subdivision of a GR with arear19 × r18 = r37 contains (a + b + c + d) = (714 + 442 + 714 + 441) GRs of areas r7, r5, r3, and r ,respectively. This corresponds to the identity φ(37−1)/2 = φ18 = 714φ3 +442φ2 +714φ+441, whichsimplifies (using φ2 = φ + 1) to φ18 = 2584φ + 1597. That in turn is an instance of the generalidentity φn = Fnφ + Fn−1, in this case using the Fibonacci number pair (F18,F17) = (2584,1597).3

The emergent global structures in these self-tilings are also unexpected, beautiful, intriguing,and can be remarkably different from one another. But because of the much more homogeneouscollection of tile sizes, it becomes quite difficult to perceive the original A, B, and C, or otherwisediscern the rule by which tiles are placed. Figures 5 and 6 show four examples of non-periodic tilingpatterns (on their sides) exhibiting large-scale structures, both “regular” and seemingly irregular.

A3 B

2 C

0

A0 B

2 C

1

Figure 5: Two lag subdivisions, using symmetry A3 B2 C0 (left) and A0 B2 C1 (right).

3It’s an interesting exercise to see what happens when you run n backwards for n < 0 in φn = Fnφ + Fn−1.

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Especially intriguing about the two tilings in Figure 6 is that the large-scale emergent lines—whether diagonal or horizontal/vertical—occur in either pairs or as singletons, depending on thewidth of the gap between them. A relationship to the distribution of 0s and 1s in the binaryFibonacci word (sequence A003849 in [5]) would appear to be an excellent conjecture: the tree oflinear transformations and subdivisions are likely isomorphic to the same production rule system.Figure 6, right, which I like to call a “Golden Plaid,” exhibits the same structure horizontally asvertically. The stripes in one direction are larger than the stripes in the other by a factor of r =

√φ.

A0 B

3 C

0

A0 B

2 C

3

Figure 6: Two more self-tilings. Left: Symmetry A0 B3 C0. Right: Symmetry A0 B2 C3 (a golden plaid!)

Ammann’s Aperiodic Golden Bee Tile

After carving GR up into A, B, and C, remove C. The result is also a self-similar figure that can bedivided into just two different-sized copies of itself, each a mirror of the other, as in Figure 7, right.This shape is known as Ammann’s “golden bee” tile [1]. With appropriate marking rules, just thesetwo sizes (the larger is the smaller scaled by r) can tile the plane aperiodically [2].

AB

C

r 6

r 5r 4

r 4

r 3

r 2

r 2r 3 r 5

r 1

r 0 = 1

Figure 7: Deriving Ammann’s bee tile from GR.

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In 1977, Robert Ammann had been exploring aperiodic tilings built from various shaped tileswhose edges were based on powers of

√φ. His work was first described in 1986 in Grunbaum’s

Tilings and Patterns [2]. But some years before, some of Ammann’s notes made their way viaMartin Gardner to Benoît Mandelbrot, who in 1980 asked one of his illustrators (the present author)what to make of them. This is when I became aware of how awesome the

√φ × 1 rectangle was.

Figure 8: “Golden Gnomon” shows myriad Ammann tiles, surrounding an unsubdivided GR.

After going to a talk by Ammann a few years later (before he disappeared and died in obscurity),I explored and played with the bee’s recursive subdivision. Based on techniques similar to thosedescribed above, an early mathematical art piece of mine, drawn in a pre-PostScript era on a penplotter, entitled “Golden Gnomon” (Figure 8), was first exhibited at Yale University in 1985 [4].

Locally largest sub-tiles form diagonal “lines” of bee tiles along, e.g., the upper-left to lower-right diagonal of GR, and those line’s similar analogs at smaller scales. By virtue of local symme-tries, these lines implicitly create ghostly oval shapes of many sizes, creating a fascinating, almosttie-dye-like texture to the image. And, perhaps unsurprisingly, one can also find the Fibonaccinumbers as an emergent property.

Looking at Figure 8, consider the set of horizontal bands, where the topmost band is boundedabove by the line along the top of GR, and below by the line that passes along the bottom of thesmaller unsubdivided GR (in the upper right of GR). These lines appear to cleave GR into two

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rectangles, but when you look carefully, they do not at locally lightest spots where a largest sub-tilefinds itself. The height of each subsequent band (counting and measuring downwards) diminishesby a factor of r2 = φ. Two diagonal lines of locally largest golden bee tiles intersect at variouspoints on the cleavage line between each band, where a locally largest Ammann tile interrupts thecleavage line otherwise dividing one band from the next. So the line across the top of the figure iscompletely straight; it has 0 interruptions. The next horizontal cleavage line down (by a factor of φ)has 1 interruption, as does the line below it. The next cleavage line below that has 2 interruptions,the next below has 3, then 5, then 8, then 13, etc.

Prayer for Relief

Call me a φnicky cynic, but the usual φ × 1 MGR plainly suffers from a golden identity crisis. It isworthy of at least 61.8033. . .% less attention than it regularly receives as an overly celebrated—yetminimally self-referential—geometric object. Whereas the more rephined

√φ × 1 rectangle is a

lean, golden mean, æsthetic φst-φghting machine. In both orthogonal directions it is sublimelyand phierarchically suffused everywhere with Goldenness and Phibonacciness, evincing far moresatisφing aesthetic sensibilities, due to the invariably intriguing combination of self-similar regu-larity, aperiodicity, and asymmetry.

So to the inφdel Miserable Golden Rectangle™, this aφcionado most conφdently shouts tophigh heaven: Φ on thee, you perphidious φgment of φine Platonic design! A new rectangle’s intown! Your days of divinity are phinitely numbered, your phistory is φnally phinished!

I, for one, root for√φ × 1. Ammann.4

Acknowledgment

The amusing portions of the foregoing owe several high-φves to theΦresign Theater, whose surreal,artful, and deφant “everything you know is wrong” aural antics and sound-of-words play long agopermanently disφgured a signiφcant “ratio” of the neural φbers within the author’s unripened brain.

References

[1] M. Barnsley and A. Vince. “Self-Similar Polygonal Tilings.” Amer. Math. Monthly,Dec. 2018, pp. 906–907.

[2] B. Grunbaum. Tilings and Patterns. W. H. Freeman, 1986, pp. 550–557.[3] P. Gailiunas. “The Golden Spiral: The Genesis of a Misunderstanding.” Bridges Conference

Proceedings. Baltimore, Maryland, USA, 2015, pp. 159–166.http://archive.bridgesmathart.org/2015/bridges2015-159.html (as of 5/1/2018).

[4] E. Sigman. “The mathemæsthetics of fractals.” Yale Daily News, Nov. 23, 1985.http://digital.library.yale.edu/utils/getarticleclippings/collection/yale-ydn/id/159176/articleId/

MODSMD_ARTICLE2/compObjId/159186/ (as of 5/1/2018).[5] N. A. J. Sloane. The On-Line Encyclopedia of Integer Sequences. https://oeis.org/A003849 (as

of 5/1/2018).

4 This essay is for humorous and mathemæsthetic purposes only. The author disavows any support for any tenet of proportionaldivinity, sacred geometry, numerology, or other faith-based attempts to impart mysterious meaning to the myriad emergentphenomena of mathematical constraints whose beauties can only be understood from a position of enlightened and ratio-nal strength.

— Caphiat Emptor —

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